Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brab2ga Structured version   Unicode version

Theorem brab2ga 4872
 Description: The law of concretion for a binary relation. See brab2a 4846 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
brab2ga.1
brab2ga.2
Assertion
Ref Expression
brab2ga
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem brab2ga
StepHypRef Expression
1 brab2ga.2 . . . 4
2 opabssxp 4871 . . . 4
31, 2eqsstri 3437 . . 3
43brel 4845 . 2
5 df-br 4367 . . . 4
61eleq2i 2498 . . . 4
75, 6bitri 252 . . 3
8 brab2ga.1 . . . 4
98opelopab2a 4678 . . 3
107, 9syl5bb 260 . 2
114, 10biadan2 646 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1872  cop 3947   class class class wbr 4366  copab 4424   cxp 4794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802 This theorem is referenced by:  fnse  6868  ltxrlt  9655  ltxr  11366  gaorb  16904  ispgp  17187  efgcpbllema  17347  lmbr  20216  isphtpc  21967  vitalilem1  22508  vitalilem2  22509  vitalilem3  22510  filnetlem1  30983
 Copyright terms: Public domain W3C validator